# The Casimir Densities for a Sphere in the Milne Universe

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## Abstract

**:**

## 1. Introduction

## 2. Geometry and the Scalar Field Modes

## 3. Region inside the Sphere

#### 3.1. Normalized Mode Functions

#### 3.2. Hadamard Function

#### 3.3. VEV of the Field Squared

#### 3.4. VEV of the Energy-Momentum Tensor

## 4. Exterior Region

#### 4.1. Scalar Modes and the Hadamard Function

#### 4.2. VEVs of the Field Squared and Energy-Momentum Tensor

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Summation Formula

## References

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**Figure 2.**The sphere-induced VEV of the field squared for $D=3$ scalar field as a function of the radial (

**left panel**) and time (

**right panel**) coordinates. The left panel is plotted for $mt=1$ and the numbers near the curves correspond to the values of the sphere radius ${r}_{0}$. For the right panel ${r}_{0}=2$ and the numbers near the curves present the values of the radial coordinate. The full and dashed curves correspond to Dirichlet and Robin (with $\beta =-0.6$) boundary conditions, respectively.

**Figure 3.**The sphere-induced contribution in the VEV of the field squared for $D=3$ scalar field versus the Robin coefficient. The graphs are plotted for $mt=1$, ${r}_{0}=2$ and the numbers near the curves correspond to the values of the radial coordinate r.

**Figure 4.**The same as in Figure 2 for the boundary-induced energy density as a function of the radial (

**left panel**) and time (

**right panel**) coordinates in the case of a minimally coupled field.

**Figure 5.**The same as in Figure 4 for the boundary-induced energy density as a function of the radial (

**left panel**) and time (

**right panel**) coordinates for a conformally coupled field.

**Figure 6.**The same as in Figure 3 for the boundary-induced contribution in the VEV of the energy density for the cases of minimally (

**left panel**) and conformally (

**right panel**) coupled fields.

**Figure 7.**The same as in Figure 4 for the energy flux density as a function of the radial (

**left panel**) and time (

**right panel**) coordinates.

**Figure 8.**The same as in Figure 7 for the energy flux density as a function of the radial (

**left panel**) and time (

**right panel**) coordinates in the case of a conformally coupled field.

**Figure 9.**The same as in Figure 6 for the energy flux density in the cases of minimally (

**left panel**) and conformally (

**right panel**) coupled fields.

**Figure 10.**The boundary-induced contribution in the VEV of the field squared outside a sphere for $D=3$ scalar field as a function of the radial coordinate. The graphs are plotted for $mt=1$, ${r}_{0}=1,1.5,2$ (numbers near the curves).

**Figure 11.**The boundary-induced contributions in the VEV of the energy density outside the sphere for minimally (

**left panel**) and conformally (

**right panel**) coupled fields. The graphs are plotted for $D=3$, $mt=1$, and ${r}_{0}=1,1.5,2$ (numbers near the curves).

**Figure 12.**The same as in Figure 11 for the energy flux density in the cases of minimally (

**left panel**) and conformally (

**right panel**) coupled fields.

${\mathit{r}}_{0}$ | 0.5 | 1 | 1.5 | 2 | 4 | 6 | 8 | 10 |
---|---|---|---|---|---|---|---|---|

$-{\beta}_{0}^{\left(\mathrm{i}\right)}\left({u}_{0}\right)$ | 0.164 | 0.313 | 0.438 | 0.537 | 0.751 | 0.833 | 0.875 | 0.9 |

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Saharian, A.A.; Petrosyan, T.A.
The Casimir Densities for a Sphere in the Milne Universe. *Symmetry* **2020**, *12*, 619.
https://doi.org/10.3390/sym12040619

**AMA Style**

Saharian AA, Petrosyan TA.
The Casimir Densities for a Sphere in the Milne Universe. *Symmetry*. 2020; 12(4):619.
https://doi.org/10.3390/sym12040619

**Chicago/Turabian Style**

Saharian, Aram A., and Tigran A. Petrosyan.
2020. "The Casimir Densities for a Sphere in the Milne Universe" *Symmetry* 12, no. 4: 619.
https://doi.org/10.3390/sym12040619